UDC 517.512+519.21

MATHEMATICS

V. V. PETROV

ESTIMATING THE CLOSENESS OF FUNCTIONS OF BOUNDED VARIATION FROM THE CLOSENESS OF THEIR FOURIER–STIELTJES TRANSFORMS

(Presented by Academician Yu. V. Linnik on 1 XII 1969)

An important role in probability theory is played by Esseen’s inequalities \((^{1,2})\), which make it possible to estimate the closeness of a nondecreasing bounded function \(F(x)\) and a function of bounded variation \(G(x)\) from the closeness of the corresponding Fourier–Stieltjes transforms on some finite interval. It is assumed here that either \(G(x)\) has a derivative uniformly bounded in \(x\), or \(F(x)\) is a purely discontinuous function such that the functions \(F(x)\) and \(G(x)\) may have discontinuities only at points \(x=x_\nu\) \((x_\nu<x_{\nu+1}, \nu=0,\pm1,\pm2,\ldots)\), satisfying the condition \(\min_\nu (x_{\nu+1}-x_\nu)\ge L>0\), with \(|G'(x)|\le A\) everywhere, except at the points \(x=x_\nu\). These results of Esseen can be encompassed by a single formulation. The following theorem is a generalization of Esseen’s theorems.

Theorem 1. Let \(F(x)\) be a nondecreasing function, and \(G(x)\) a function of bounded variation on the real line, \(F(-\infty)=G(-\infty)\), \(F(+\infty)=G(+\infty)\). Let

\[ f(t)=\int_{-\infty}^{\infty} e^{itx}\,dF(x), \qquad g(t)=\int_{-\infty}^{\infty} e^{itx}\,dG(x), \]

and let \(T\) be an arbitrary positive number. Then, for any number \(b>1/2\pi\), the inequality

\[ \sup_x |F(x)-G(x)| \le b\int_{-T}^{T}\left|\frac{f(t)-g(t)}{t}\right|\,dt + bT\sup_x \int_{|y|\le c(b)/T}|G(x+y)-G(x)|\,dy, \tag{1} \]

holds, where \(c(b)\) is a positive constant depending only on \(b\).

In inequality (1) one may take \(c(b)\) equal to the root of the equation

\[ \int_0^{\,1/4\,c(b)} \frac{\sin^2 u}{u^2}\,du = \frac{\pi}{4}+\frac{1}{8b}. \]

In the special case when \(F(x)\) and \(G(x)\) are distribution functions, a similar result was obtained by A. C. Feinleib \((^3)\). A uniform estimate of the difference between a distribution function \(F(x)\) and a certain function of bounded variation \(G(x)\) that is not a distribution function is of considerable interest for applications (for example, in the study of asymptotic expansions in limit theorems for sums of independent random variables).

If the conditions of Theorem 1 are satisfied and the function \(G(x)\) satisfies the following Lipschitz condition:

\[ |G(x)-G(y)|\le K|x-y|^\alpha \]

for all \(x\) and \(y\) and for some positive constants \(K\) and \(\alpha\), then the second term on the right-hand side of inequality (1) may be replaced by \(2bK(c(b))^{1+\alpha}(1+\alpha)^{-1}T^{-\alpha}\).

We give one more immediate consequence of Theorem 1, which is also a generalization of Esseen’s theorem.

Theorem 2. Let \(F(x)\) be a nondecreasing function, \(G(x)\) a function of bounded variation, and let \(f(t)\) and \(g(t)\) be the corresponding Fourier–Stieltjes transforms; let \(T\) be an arbitrary positive number, and
\[ F(-\infty)=G(-\infty), \qquad F(+\infty)=G(+\infty). \]
Suppose \(|G'(x)| \le A\) everywhere, except at points of discontinuity of the function \(G(x)\).

Then, for any number \(b>1/2\pi\), the inequality
\[ \sup_x |F(x)-G(x)| \le b \int_{-T}^{T} \left|\frac{f(t)-g(t)}{t}\right|\,dt + r(b)\frac{A}{T}, \tag{2} \]
holds, where \(r(b)\) is a positive constant depending only on \(b\).

In inequality (2) one may take \(r(b)=bc^2(b)\), where \(c(b)\) is the constant from Theorem 1.

Leningrad State University
named after A. A. Zhdanov

Received
1 XII 1969

REFERENCES

\({}^{1}\) C. G. Esseen, Acta Math., 77, 1 (1945).
\({}^{2}\) B. V. Gnedenko, A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Moscow–Leningrad, 1949.
\({}^{3}\) A. C. Feinleib, Izv. Acad. Sci. USSR, Ser. Math., 32, No. 4, 859 (1968).